# Piotr Liczberski's research while affiliated with Lodz University of Technology and other places

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## Publications (54)

It is known that the starlikeness plays a central role in complex analysis, similarly as the convexity in functional analysis. However, if we consider the biholomorphisms between domains in $${\mathbb {C}}^{n},$$ C n , apart from starlikeness of domains, various symmetries are also important. This follows from the Poincaré theorem showing that the...

The paper is devoted to the study of a family of complex-valued holomorphic functions and a family of holomorphic mappings in $${\mathbb {C}}^{n}.$$ C n . More precisely, the article concerns a Bavrin’s family of functions defined on a bounded complete n -circular domain $${\mathcal {G}}$$ G of $${\mathbb {C}}^{n}$$ C n and a family of biholomorphi...

In the [1], [4], [3] and [2] there were examined the Bavrin?s families (of holomorphic functions on bounded complete n? circular domains G ?Cn) in which the Temljakov operator Lf was presented as a product of a holomorphic function h with a positive real part and the (0, k)?symmetrical part of the function f,(k ? 2 is a positive integer). In [17] t...

In the paper there is considered a generalization of the well-known Fekete–Szegö type problem onto some Bavrin’s families of complex valued holomorphic functions of several variables. The definitions of Bavrin’s families correspond to geometric properties of univalent functions of a complex variable, like as starlikeness and convexity. First of all...

This paper is devoted to a generalization of the well-known Fekete-Szegö type coefficients problem for holomorphic functions of a complex variable onto holomorphic functions of several variables. The considerations concern three families of such functions f, which are bounded, having positive real part and which Temljakov transform Lf has positive...

The paper concerns holomorphic functions in complete bounded n-circular domains \({{\mathcal {G}}}\) of the space \({\mathbb {C}}^n\). The object of the present investigation is to solve majorization problem of Temljakov operator. This type of problem has been studied earlier in Liczberski and Żywień (Folia Sci Univ Tech Res 33:37–42, 1986), Liczbe...

The paper is devoted to the investigations of holomorphic functions on complete n-circular domains G of ℂn which are solutions of some partial differential equations in G. Our considerations concern a collection M
Gk, k ≥ 2, of holomorphic solutions of equations corresponding to planar Sakaguchi’s conditions for starlikeness with respect to k-symme...

The paper concerns investigations of holomorphic functions of several complex variables with a factorization of their Temljakov transform. Firstly, there were considered some inclusions between the families \(\mathcal {C}_{\mathcal {G}},\mathcal {M}_{\mathcal {G}},\mathcal {N}_{\mathcal {G}},\mathcal {R}_{\mathcal {G}},\mathcal {V}_{\mathcal {G}}\)...

In this paper, we will consider classes of subordinations involving partial derivatives of holomorphic mappings in complex Banach spaces.

In this paper, we will consider some sufficient conditions for locally biholomorphic mappings defined in the unit ball in complex Banach spaces to be biholomorphic and to have Φ-like images. As a corollary, we obtain some sufficient condition for locally biholomorphic mappings to be starlike mappings.

The subject of the paper concerns the existence of different inclusions between families St, Sc of C-n - biholomorphic starlike or convex mappings and a family S(k), k >= 2, of C-n - locally biholomorphic mappings. In the definition of S(k) we used a combination of a Kikuchi-Matsuno-Snifridge's starlikeness condition and a property of (j, k)-symmet...

In this paper some characterizations and properties of domains in Re with boundary accessible externally by some circular cones are given. It is shown that the boundary of such domains is also conically accessible from the interior. It is also characterized alpha-accessible domains of C-N, which are biholomorphically equivalent to the open Euclidea...

We continue a concept of strongly starlike mappings of order alpha on bounded balanced pseudoconvex domains with C 1 -plurisubharmonic defining function, introduced by H. Hamada, G. Kohr and M. Kohr [Rev. Roum. Math. Pures Appl. 44, No. 4, 583–594 (1999; Zbl 0999.32008)]. The main theorem contains two new geometric properties of such mappings and g...

In the paper an internal geometric characterization of strongly starlike mappings of order alpha in CnCn is given.

In the paper necessary and sufficient conditions for the boundary starlikeness of holomorphic mappings in Cn are given. In the proof an n-dimensional version of Julia's theorem is used.

The paper concerns holomorphic functions in complete bounded n-circular domains of the space ℂⁿ and presents some properties of the above mentioned functions belonging to the families described by some geometrical or analytical conditions. This subject has been considered by many mathematicans, for example I.I. Bavrin, K. Dobrowolska, I. Dziubiński...

In this paper we offer new results of research presented in the referred papers of J.E. Fornaess and E.L. Stout, E. Ligocka and the authors, and concerning the existence of m-valent locally biholomorphic mappings from product domains of Cn onto n-dimensional complex manifolds. In particular, we confirm an own conjecture about the estimation of the...

We continue E. Ligocka’s [Ann. Pol. Math. 82, No. 2, 127–135 (2003; Zbl 1057.30019)] investigations concerning the existence of m-valent locally biholomorphic mappings from multi-connected onto simply connected domains. We decrease the constant m, and also give the minimum of m in the case of mappings from a wide class of domains onto the complex p...

In this paper, we consider the problem of distortion theorems for mappings which map the unit ball biholomorphically onto
convex domains in ℂ
n
. In particular, we discuss two distortion conjectures for such mappings.

In the paper there are given new applications of dimen- sion reduction method (from (3)) of studing linearly invariant families of holomorphic mappings in n. There also included generalizing mod- ications of this method.

In the paper the decomposition with respect to the group of roots of unity is applied to characterize some classes of biholomorphic mappings in There is considered the problem of subordination and majorization relations between convex mappings and their components in the above partition. There is also given a distortion theorem for convex mappings...

In the paper the problem of sharp lower estimation for ‖Df(z)‖ in the class of normalized biholomorphic mappings f between the open unit ball Bn and convex domains in Cn has been considered.

In the present paper we give a method of obtaining some solutions for functional equations in which the unknown function occurs in the form of its own iterates. We reduce the equation to one of the type we have solved in [2] using properties of (j, k)-symmetrical functions, which are collected in [1].

We continue our previous work on a problem of Janiec connected with a uniqueness theorem, of Cartan{Gutzmer type, for holomorphic mappings in C n . To solve this problem we apply properties of (j;k)-symmetrical functions.

The notion of a linearly invariant family of mappings of a ball in C was introduced in the article ``Pfaltzgraff J. A., Distortion of locally biholomorphic maps of the n-ball, Complex Variables, 33, 239-253 (1997)''''; it generalizes the classical case n=1 studied earlier by Ch. Pommerenke and other authors. In the indicated article, Pfaltzgraff in...

In this paper, we will give the growth theorem of starlike mappings of order on the unit ball B in complex Banach spaces. We also give an analytic sucien t condition for a locally biholomorphic mapping on B to be a starlike mapping of order .

In this paper we suggest a new definition of the order of a linearly invariant family of locally biholomorphic mappings of the unit ball in . This definition is equivalent to the one given by Pfaltzgraff in [J.A. PfaltzgraffComplex Variables Theory Appl 33 (1997), 239–253.]. It bases on a very simple relationship with the Jacobian of the mappings (...

In a recent paper, Gurganus extended Brickman's work on Φ-like holomorphic functions on the unit disc in C to locally biholomorphic mappings on the unit ball in a Banach space. In this paper, we will extend the above results to locally biholomorphic mappings on bounded balanced pseudoconvex domains with C 1 plurisubharmonic defining functions in C...

In this paper we introduce the concept of strongly starlikeness of order α > 0, for holomorphic mappings defined on the unit ball of Cn. We obtain the distorsion and the covering theorems for strongly starlike mappings of order α ∈ (0,1] and we give a connection between strongly starlikeness and spirallikeness in Cn. http://web.math.hr/glasnik/vol_...

Introduction. In the present paper the authors study some families of functions from a complex linear space X into a complex linear space Y . They introduce the notion of (j, k)-symmetrical function (k = 2, 3, . . .; j = 0, 1, . . . , k - 1) which is a generalization of the notions of even, odd and k-symmetrical functions. It has turned out that fo...

P. T. Mocanu in the paper [3] has shown that the well known univalence criterions for holomorphic functions of one variable, due to W. Alexander W. Kaplan, K. Noshiro, S. Ozaki, S. E. Warschawski and J. Wolf (see [1], [2], [4], [5], [8], [9]) can be extended onto the complex functions from the class C . The same way, in n-dimensional case, we gener...

We give a version of the distortion theorem for biholomorphic mappings between the unit polydisc and convex domains in the space of n complex variables.

The author gives a simple starlikeness criterion for locally biholomorphic mapping from the open unit ball in C n into C n.

In the recent paper [ibid. 103, Mat. Fiz. 16,. Mat. 12, 103-113 (1992)] we introduced a generalization of the notions of even functions and odd functions, proved several properties of these notions and gave its different applications. In the present paper we present several new applications of these notions for the investigation of the solutions of...

In this paper we introduce the concept of strongly starlikeness of order Ct. > 0, for holomorphic mappings defined on the unit ball of en, We obtain the distortion and the covering theorems for strongly starlike mappings of order Ct. E (0, 1) and we give a connection between strongly starlikeness and spirallikeness in en, j=1 Z E en. The open Eucli...

In this paper, we will give the growth theorem of starlike mappings of order α on the unit ball B in complex Banach spaces. We also give an analytic sufficient condition for a locally biholomorphic mapping on B to be a starlike mapping of order α. http://web.math.hr/glasnik/vol_36/no1_05.html

## Citations

... In Xu and Liu [23], the Fekete-Szegö inequality for starlike mappings in several complex variables was first obtained. Very recent important results related to the Fekete-Szegö inequality in several complex variables were obtained in other articles (see, e.g., Długosz and Liczberski [24], Elin and Jacobzon [25], Hamada [26], and Lai and Xu [27]). In particular, the Fekete-Szegö inequality for univalent mappings in several complex variables was first obtained in Hamada et al. [28]. ...

... In particular, the Fekete-Szegö inequality for univalent mappings in several complex variables was first obtained in Hamada et al. [28]. Also, the other related results may consult in Długosz and Liczberski [29], Graham and Kohr [30], and Nunokawa and Sokol [31]. Liu et al. [32] considered only the main coefficients that are analogous with the diagonal elements of a square matrix and generalized Theorem A to the case on a Reinhardt domain in n from a new viewpoint. ...

... In the papers [4,5] we gave for a few Bavrin's families a sharp estimate for the pair of homogeneous polynomials Q f ,2 , Q f ,1 , i.e., the sharp estimate ...

... The theory of functions exhibiting (x, y)-symmetry has a wide range of intriguing applications. For instance, these functions are useful in exploring the set of fixed points of mappings, estimating the absolute value of certain integrals, and deriving results akin to Cartan's uniqueness theorem for holomorphic mappings, as demonstrated in [1]. The intrinsic properties of (x, y)-symmetrical functions are of great interest in the field of Geometric Function Theory. ...

... However, Cartan [6] stated that the Bieberbach conjecture does not hold in several complex variables. In fact, only a few complete results are known for the inequalities of homogeneous expansions for subclasses of biholomorphic mappings in n (see, e.g., Długosz and Liczberski [7], Hamada and Honda [8], Liu and Wu [9], Liu and Liu [10], Liu et al. [11], and Xu et al. [12]). Many works are concentrating on the bounds of second-and third-order terms of homogeneous expansions for starlike mappings and the sharp bounds of all homogeneous expansions for the special subclasses of starlike mappings with some restricted conditions (see, e.g., Hamada et al. [13], Liu and Liu [14], Tu and Xiong [15], Xiong [16], Xu et al. [17], and Xu et al. [18]). ...

... For a characterization of starlike mappings G of the form (3.4), see, e.g., [7,13,24,28,33]. For instance in the paper [28] we can found the following result: ...

... In the proof of next theorem, we will use the following result form the paper [10]: ...

... Continuing the first section we give a very useful functional symmetry. In the papers [3,6,11] there are considered the consequences of a modification of the above starlikeness factorization (1), using a unique decomposition of mappings f : B n → C n with respect to the cyclic group of k-th. roots of unity, k ≥ 2. Below we present such partition for mappings f : X ⊃ → Y, where X, Y are normed complex vector spaces and is a k-symmetric nonempty subset of X (ε = for the generator ε = exp 2πi k of the above group) [14]. ...

... In [26] the notion of α-accessible domain was defined. It was shown that such domains posses the cone condition with l(p) = −p. ...

... z -h(z) for zeB.Ii we show that R e m a r k 3. In [7], the second and third authors gave sufficient conditions for locally biholomorphic mappings on the unit ball with respect to an arbitrary norm on C n to be $-like mappings. The condition (i) is the same, but the right hand side of the condition (ii) in [7] is 1. ...